2 Answers
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A major list of incomplete things in M-theory (I will ignore those that are not M-theory specific, which you can find in generic string theory questions):

2-brane constructions--- if you make a 2-brane stack, this is described by Schwarz Chern-Simons theory, under certain conditions. Making the general AdS/CFT correspondence for the 2-brane case is important, and I believe still unsolved.

5-brane constructions--- what is the AdS/CFT for five-branes? We don't know as far as I know.

What are the complete orbifold types for M-theory--- for strings, the orbifolding was understood from wordsheet considerations, but in M-theory, the Horava-Witten domain wall was understood from anomaly cancellation and needing to incorporate heterotic strings. Are these constructions fully consistent in M-theory (beyond string theory) (I am certain that yes)? What other orbifolded theories have we missed?

What is the relation between orbifolds and branes? Can one resolve orbifolds into combinations of matter branes?

What are orbifold dynamics--- they don't jiggle like normal gravitational things.

At what dimension does maximal supergravity stop being renormalizable?

Is there a complete intrinsic description of M-theory which is not based on a IIA theory compactification (Matrix theory), or on a near-2brane or near-5brane description?

I don't know if all of these are still active, or still unsolved, but these are the things that worry me when I think about M-theory personally. There is another thing that is not M-theory specific, but where I think M-theory methods are most promising:

What are black hole emissions--- unitarity demands that what information goes in to a black hole comes out. What comes out of black holes? Is it always thermal? I believe not, and the issue will be resolved soon.

@Dimension10: Matrix theory can reproduce M5 branes, opaquely. I just mean that there is a little-string theory living on 5-branes which, on general AdS/CFT principles, is supposed to be dual to the gravitational theory near the branes, and it is ill understood what the theory is in M-theory, as opposed to type II string theory, or how the duality works properly. It's something that was just barely sorted out for the M2 branes, and it is useful for knowing the stuff that's consistent in M-theory beyond the classical supergravity limit, we don't have strings to probe M-theory with, only branes.
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Ron MaimonJul 27 '13 at 4:33

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@Dimension10: if you look at Maldacena's original paper arxiv.org/abs/hep-th/9711200 he lists three basic examples of AdS/CFT duality. In each case there is a stack of branes with an AdS geometry in the bulk near the branes, and it's string/M theory in that space, which is equivalent to a superconformal theory which is the worldvolume theory of the brane stack...
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Mitchell PorterAug 2 '13 at 9:30

One basic thing that's incomplete about M-Theory is that the theory isn't even completely defined. One can talk about M-Theory in certain limits where it reduces to string theory, or one can talk about the low energy (non-quantum) supergravity theory which M-Theory reduces to, but there is no definition of what M-Theory actually is as a quantum theory in 11 non-compact dimensions. In particular, the full quantum mechanical lagrangian for M2 branes is not known, although classical limits may be deduced.

@dimension10: Not completely, because it's defined around a flat background. All the solutions are in a fixed background, either AdS or flat, or some other supersymmetric background. You can't describe deSitter spaces, or backgrounds that interpolate between different finite-energy-density states, like Simeon Hellerman's 26 dimensional bosonic string cosmology, going to a standard string theory. Is it completely consistent non-perturbatively? That's what a non-perturbative formulation should answer. The Matrix theory and AdS/CFT are only valid as finite-number-of-particles over solvable vacuum
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Ron MaimonJul 27 '13 at 18:01

@RonMaimon: Doesn't knowing it for AdS space help the understanding for other spaces too? I thought it was already done... Does that mean that even the Lag. Density is undefined or just some of the results, like the scattering amplitudes, etc. ?
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Dimensio1n0Jul 27 '13 at 18:11

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There's a lot more to do, but it seems people are pretty stuck at the moment. The last major breakthrough was the M2 brane CFT by Schwartz and co, and I haven't followed the last few years. The Lagrangian is not exactly what you are after, you want to know the scattering amplitudes--- it's not from a local Lagrangian density in string theory. The scattering amplitudes are only well defined in certain backgrounds, where there is a known CFT dual. The general formulation is lacking, and it probably requires a major new idea.
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Ron MaimonJul 28 '13 at 10:15